Physics 101: Lecture 20 Elasticity and Oscillations

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Physics 101 Lecture 20 Elasticity and
Oscillations

• Todays lecture will cover Textbook Chapter
  10.5-10.10

I have absolutely no idea. I have been studying
for the exam tonight since last sunday, and all
day today. I came home from the exam only to
slave over the physics homework until now, and
now I'm doing this preflight. I am on physics
OVERLOAD and going cross eyed. I seriously think
that it is cruel and unfair that we have a
homework due after a test

• Hour exam II average 84.6
• Most common (scaled) score 100 (81 or 13)
• In the interest of fairness, the next one will be
  harder.

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Review Energy in SHM

• A mass is attached to a spring and set to motion.
  The maximum displacement is xA
• Energy U K constant!
• ½ k x2 ½ m v2
• At maximum displacement xA, v 0
• Energy ½ k A2 0
• At zero displacement x 0
• Energy 0 ½ mvm2
• ½ k A2 ½ m vm2
• vm ?(k/m) A
• Analogy with gravity/ball

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Kinetic Energy ACT

• In Case 1 a mass on a spring oscillates back and
  forth. In Case 2, the mass is doubled but the
  spring and the amplitude of the oscillation are
  the same as in Case 1. In which case is the
  maximum kinetic energy of the mass bigger?
• A. Case 1
• B. Case 2
• C. Same

½kA2 ½mvm2
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Potential Energy ACT

• In Case 1 a mass on a spring oscillates back and
  forth. In Case 2, the mass is doubled but the
  spring and the amplitude of the oscillation are
  the same as in Case 1. In which case is the
  maximum potential energy of the mass and spring
  bigger?
• A. Case 1
• B. Case 2
• C. Same

Maximum displacement x A Energy ½ k A2 0
Same for both!
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5
Velocity ACT

• In Case 1 a mass on a spring oscillates back and
  forth. In Case 2, the mass is doubled but the
  spring and the amplitude of the oscillation are
  the same as in Case 1. Which case has the larger
  maximum velocity?
• 1. Case 12. Case 23. Same

Same maximum Kinetic Energy K ½ m v2
smaller mass requires larger v
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Simple Harmonic MotionQuick Review
x(t) Acos(?t) v(t) -A?sin(?t) a(t)
-A?2cos(?t)
x(t) Asin(?t) v(t) A?cos(?t) a(t)
-A?2sin(?t)
OR
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency ?
2?f 2?/T
xmax A vmax A? amax A?2
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Natural Period T of a Spring

• Simple Harmonic Oscillator
• w 2 p f 2 p / T
• x(t) A cos(wt)
• v(t) -Aw sin(wt)
• a(t) -Aw2 cos(wt)
• Draw FBD write Fma
• -k x m a
• -k A m amax
• -k A m (-A w2)
• Aw2 (k/m) A
• w ?(k/m)

A,m,k dependence demo
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Period ACT

• If the amplitude of the oscillation (same block
  and same spring) is doubled, how would the period
  of the oscillation change? (The period is the
  time it takes to make one complete oscillation)
• A. The period of the oscillation would double.B.
  The period of the oscillation would be halvedC.
  The period of the oscillation would stay the same

x
2A
t
-2A
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Equilibrium position and gravity

• If we include gravity, there are two forces
  acting on mass. With mass, new equilibrium
  position has spring stretched d
• SFy 0
• kd mg 0
• d mg/k Let this point be y0
• SF ma
• k(d-y) mg ma
• -k y ma
• Same as horizontal! SHO
• New equilibrium position y0
• corresponds to height -d

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Vertical Spring ACT

• Two springs with the same k but different
  equilibrium positions are stretched the same
  distance d and then released. Which would have
  the larger maximum kinetic energy?
• 1) M 2) 2M 3) Same

Y0
Just before being released, v0 yd Etot 0 ½
k d2 Same total energy for both When pass
through equilibrium all of this energy will be
kinetic energy again - same for both!
Y0
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Pendulum Motion
Bowling ball pendulum

• For small angles
• T mg cos(q) mg
• Tx -mg sin(q) -mg x/L Note F
  proportional to x!
• S Fx m ax
• -mg (x/L) m ax
• ax -(g/L) x
• Recall for SHO a - w2 x
• w ?(g/L)
• T 2 p ?(L/g)
• Period does not depend on A, or m!

q
L
T
m
x
mg
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Preflight 1

• Suppose a grandfather clock (a simple pendulum)
  runs slow. In order to make it run on time you
  should
• 1. Make the pendulum shorter
• 2. Make the pendulum longer

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Elevator ACT

• A pendulum is hanging vertically from the ceiling
  of an elevator. Initially the elevator is at
  rest and the period of the pendulum is T. Now
  the pendulum accelerates upward. The period of
  the pendulum will now be
• A. greater than T
• B. equal to T
• C. less than T

g is effectively bigger, T is lower.
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Preflight

• Imagine you have been kidnapped by space invaders
  and are being held prisoner in a room with no
  windows. All you have is a cheap digital
  wristwatch and a pair of shoes (including
  shoelaces of known length). Explain how you might
  figure out whether this room is on the earth or
  on the moon

Since you know the length of the shoelace then
you can build a pendulum and see how long it
takes for one period. If its longer than what you
expect on earth than you must be on the moon, if
its the same you're on earth.
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Alien Preflight
You could drop a shoe from a known height and
also measure the time it takes for the shoe to
hit the ground. Then you can use xxo vot
1/2at2 to find the acceleration. The shoe starts
at rest (vo0). If the acceleration is less than
9.8, you're on the moon.
I could jump. that would be enough really. The
gravitational difference is quite significant
between the two. but I suppose I could make a
pendulum of the watch and shoelace if I wanted to
do extra work and was bored.
I have absolutely no idea. I have been studying
for the exam tonight since last sunday, and all
day today. I came home from the exam only to
slave over the physics homework until now, and
now I'm doing this preflight. I am on physics
OVERLOAD and going cross eyed. I seriously think
that it is cruel and unfair that we have a
homework due after a test. As for this pendulum,
I think that you could tie a shoe to a shoelace
and start it swinging in a pendulum motion. The
period of the pendulum (T) could be approximated
by 2Pi square root (length of shoelaces/g). You
have your watch to time the pendulum, and so you
could determine if g is the acceleration of the
Earth (9.8) or a martian planet!
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Alien Preflight
First you tie both shoe laces together and
attach that to the shoe. this will give you a
good amount of torque because the radius is
longer. then you pretend to have a heart attack.
After a while, the aliens will most likely come
and check on you. once they do, get up, start
swinging the shoe lace/shoe combo and hit one of
them in the face. if there is another one, throw
your other shoe at him. if there are even more,
headbutt them into oblivion. continue headbutting
into oblivion until all the aliens are down. find
the main control center and pilot the space ship
to the nearest pawn shop and sell the space ship
for one gazillion dollars and 33 cents. get 33
cents in pennies and start throwing them at the
aliens until they tell you were they took you.
the end. or since small oscillations depend on
gravity, you can tie the shoe laces to the shoe.
then figure out what 2pisqrt(L/9.8) is and then
swing the 'pendulum' and see if the period for
that is the same.
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Summary

• Simple Harmonic Motion
• Occurs when have linear restoring force F -kx
• x(t) A cos(wt)
• v(t) -Aw sin(wt)
• a(t) -Aw2 cos(wt)
• Springs
• F -kx
• U ½ k x2
• w sqrt(k/m)
• Pendulum (Small oscillations)
• w ?(g/L) incorrect in lecture notes

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